Why be accurate when you can have trends
"The purpose of computation is insight, not numbers.",
Richard Hamming, Numerical Methods for Scientists and Engineers, 1962.
Suppose you calculated, or measured, the lift coefficent slope of long thin
wings, and you found that CL/alpha = 6.3, 6.28, 6.283, 6.2832 for increasing
wing lengths.
A fair guess would be that, based on the trend, the true lift coefficient slope
for infinitely long wings was exactly 2*pi.
That guess might help you to prove mathematically that it is indeed 2*pi, and
that would be an excellent result.
Now, suppose that you computed an approximation of the lift coefficient slope of
a thin wing of circular planform in potential flow as 1.79075033.
Plugging that value into the so-called PSLQ algorithm in Wolfram Alpha gives the
result 16/(3*zeta(2) +4), where zeta(2) = pi^2/6.
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After a bit of algebra, that means 1.79075033 is approximately 32/(pi^2 + 8).
That might give you a hint of how to get the exact value for the lift slope. You
would still have to come up with a way to prove it. Hauptman and Miloh found
the result CL/alpha = 32/(pi^2 + 8) which is a quite lovely result IMO. They
most definitely did NOT use the PSLQ algorithm to get a "hint". I have just used
it as an example because the PSLQ algorithm found their result very easily.
Unfortunately, Hauptman and Miloh's result is not exact. They had to twist the
wingtips slightly, so their wing is not perfectly flat. It is pretty good as an
approximation, being about 0.04% different to Jordan's result. Unfortunately,
their moment coefficient slope is not as good and nor is the induced drag or the
strength of the (composite) wing tip singularity strength.
Also unfortunately, my 20 decimal place result for the lift slope doesn't yield
anything useful using PSLQ. I'm actually not that interested in the lift, moment
or induced drag - I'm more interested in the wing tip singularity strength
because it would be relevant to all wings with (parabolic) curved wing tips.
Did you ever do any drag calculations on the circular wing? There's a current thread here dealing with low aspect ratio foils where that might be relevant:
I'm hoping an experienced hand at foil design can pitch in here and check my theory. I'm currently pioneering a small hydrofoil that is working and I'm refining the foils to get lower drag as I refine my knowledge.
I'm entirely self taught and started this early December.
The issue I'm having is the xflir5 charts are telling me I could optimise the drag by increasing the angle of attack at the tips. Convention seems to be to reduce it. Along with increased AoA at the tips I only need an aspect ratio of about 5 for the amount of lift I need if I do an elliptical wing and this way.
2...
My own dabbling with circular wings was part of a validation process for my personal VL code and I think it came through with flying colors there, but I can't find any notes or results indicating that I computed the induced drag.
Your result for the lift coeffcient slope is very good, but I'm not convinced
that VLM is particularly robust. I don't have time to search for it, but I think
there is a very old thread here on bdn where Mark Drela also showed a good
approximation for the circular wing using his VLM.
The problem I see is that you can adjust the number of chordwise and spanwise
panels to get a good approximation. If you then vary one or both slightly, the
accuracy can get significantly worse. You used 32 x 32 panels. Have you tried,
for example, 30 x 34? Or 34 x 30?
Lift is an integrated quantity, and that can mask some "wobbles" in the chord-
wise loadings. I have attached David Standingford's PhD thesis. See Chapters 1
and 2.
As to induced drag...
There are at least 2 good ways to calculate induced drag. Drela and many others
prefer a Trefftz plane method. See, for example,
For panel methods, I prefer the method described by Standingford which uses the
leading edge suction because I am interested in the way it varies as you go
towards the wing tips. It is also the method I used in my program LSP which I
released on bdn many years ago.
But for the circular wing I like Jordan's method. See the attached (very)
unfinished internal report. The plots compare different extrapolation methods I
have tried.
Sorry, but some entries are missing from the tables. However, the list of
references at the end are a good start for anyone interested in this type of
extremely esoteric stuff. I will return to this work one day, hopefully before I
am taken off to the Home for the Befuddled.
Leo.
